Solve by Completing the Square x^2-2x-3/4=0

$x^{2} - 2 x - \frac{3}{4} = 0$

Add $\frac{3}{4}$ to both sides of the equation.

$x^{2} - 2 x = \frac{3}{4}$

To create a trinomial square on the left side of the equation, find a value that is equal to the square of half of $b$ .

${(\frac{b}{2})}^{2} = {(- 1)}^{2}$

Add the term to each side of the equation.

$x^{2} - 2 x + {(- 1)}^{2} = \frac{3}{4} + {(- 1)}^{2}$

Simplify the equation.

Simplify the left side.

Raise $- 1$ to the power of $2$ .

$x^{2} - 2 x + 1 = \frac{3}{4} + {(- 1)}^{2}$

Simplify the right side.

Simplify $\frac{3}{4} + {(- 1)}^{2}$ .

Raise $- 1$ to the power of $2$ .

$x^{2} - 2 x + 1 = \frac{3}{4} + 1$

Write $1$ as a fraction with a common denominator.

$x^{2} - 2 x + 1 = \frac{3}{4} + \frac{4}{4}$

Combine the numerators over the common denominator.

$x^{2} - 2 x + 1 = \frac{3 + 4}{4}$

Add $3$ and $4$ .

$x^{2} - 2 x + 1 = \frac{7}{4}$

Factor the perfect trinomial square into ${(x - 1)}^{2}$ .

${(x - 1)}^{2} = \frac{7}{4}$

Solve the equation for $x$ .

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$x - 1 = \pm \sqrt{\frac{7}{4}}$

Simplify $\pm \sqrt{\frac{7}{4}}$ .

Rewrite $\sqrt{\frac{7}{4}}$ as $\frac{\sqrt{7}}{\sqrt{4}}$ .

$x - 1 = \pm \frac{\sqrt{7}}{\sqrt{4}}$

Simplify the denominator.

Rewrite $4$ as $2^{2}$ .

$x - 1 = \pm \frac{\sqrt{7}}{\sqrt{2^{2}}}$

Pull terms out from under the radical, assuming positive real numbers.

$x - 1 = \pm \frac{\sqrt{7}}{2}$

Add $1$ to both sides of the equation.

$x = \pm \frac{\sqrt{7}}{2} + 1$

The result can be shown in multiple forms.

Exact Form:

$x = \pm \frac{\sqrt{7}}{2} + 1$

Decimal Form:

$x = 2.32287565 \dots, - 0.32287565 \dots$